Error in backpropgation In learning parameters of multilayered neural network by back-propagation, 
is the loss function quantifying scalar or vector(squared-error of input vector) difference between current hypothesis and truth(parameter) for a set of instances, as long as the network is intended to solve the problem of classification?
 A: Since you are asking if loss function is quantifying a vector. It is better to put a bit formally. 
At the output layer, the loss function is a $R^c\rightarrow R$ mapping, where $c$ is the number of classes. For multiclassification, the target in your data need to be one-hot-coded to a vector of length $c$.  For binary classification, if standard logistic regression is used, then $c=1$, if a softmax layer is used, then $c=2$. 
The mapping is basically to sum up (or average) the errors for all classes for all data examples. For each data example, you will have this
$$Loss = \sum_{i=1}^c Error^i$$
$$ Error^i=f(y_i, \hat y_i(x, w))$$
$y_i$ is the $i$th element in the one-hot coding vector, and $\hat y_i$ is its estimate by the neural network, $f(\cdot)$ can be squared error or something else. 
My answer for your question is yes, the loss function is just summing up a vector of errors for all classes. 
A: The loss function calculates the difference between an output vector (that can have a single dimension, in which case it is a scalar), and the expected output vector (again, it can have a single dimension).

EDIT :
Vector here means an ordered list of scalars. (Furthermore, there is dependence between those scalars in the multiclass case, if they represent a probability : their sum is 1).
The neural network doesn't do dot products, it does matrix multiplication, between the input (vector, aka N*1 matrix) and a weight matrix.

Supplementary note : 
For classification, the output of typical networks is a multi dimensional vector representing the probability of each class (as predicted by the network, generally indicating the confidence of the network in its prediction). The loss function used in such cases is typically cross entropy, not euclidian distance.
There are other techniques (changing the representation of the output and/or the loss function), but this is the more standard approach.
